uniformly with respect to μ∈[0,(1-δ)⁢κ] and
x∈(0,(1-δ)⁢(2⁢κ+2⁢κ2-μ2)],
where δ again denotes an arbitrary small positive constant. For the
functions J2⁢μ, Y2⁢μ, H2⁢μ(1), and
H2⁢μ(2) see §10.2(ii), and for the env
functions associated with J2⁢μ and Y2⁢μ see
§2.8(iv).

These approximations are proved in Dunster (1989). This reference
also includes error bounds and extensions to asymptotic expansions and complex
values of x.

§13.21(iv) Large κ, Other Expansions

For a uniform asymptotic expansion in terms of Airy functions for
Wκ,μ⁡(4⁢κ⁢x) when κ is large and positive, μ
is real with |μ| bounded, and x∈[δ,∞) see
Olver (1997b, Chapter 11, Ex. 7.3). This expansion is simpler in form
than the expansions of Dunster (1989) that correspond to the
approximations given in §13.21(iii), but the conditions on
μ are more restrictive.

For asymptotic expansions having double asymptotic properties see
Skovgaard (1966).